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ABCD  is a cyclic quadrilateral, then  the vertices, A, B, C and D are on the circumference of a circle.  Let  a, b,c, and d  be the sides and  e and f be the diagonals.

1.  sum of  opposite angles  is 180 deg.

2.  a* c +  b *d = e * f    
         This  is  Ptolemy's  theorem.

3.   The area of the quadrilateral is given by BrahmaGupta's theorem.
         \Delta= \sqrt{(s-a)(s-b)(s-c)(s-d)},\ \ s=\frac{a+b+c+d}{2}
        s = semiperimeter.

4.   Angle   ABD =  angle ACD
5.    Angle   BCA = angle  BDA
6.   angle    CAD =  angle  CBD
7,    angle    BAC  =  angle  BDC

1 5 1
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